Space transformations in the study of multidimensional functions in the hydrologic sciences

In hydrological sciences we often deal with complex phenomena which take place in several dimensions, such as two-dimensional distributions of rainfall over a region or three-dimensional chemical transport within an aquifer. The aim of this paper is to show benefits obtained by use of space transformations in the solution of multidimensional problems. The approach is to transform the original space to a space of lower dimensionality, where analysis is easier and involves less computational effort. In calculations involving isotropic functions, analytically and computationally tractable expressions are available for both the space transformations and their inverses. Particular attention is paid to the frequency domain forms of the space transformations. Differential equations describing flow through porous media, structural identification of spatially distributed soil variables, and estimation and simulation of hydrologic fields are a few applications where the use of these transforms may be fruitful. Simple but useful examples are included to illustrate the methodology, and in occasion we indicate some areas of further work.